Angular momentum three coupled momenta

Jx,Jy,Jz are the three components of the total angular momentum. The term Bj Jl + Jy) is the contribution to the rotational energy, while the other two terms contribute to the bending modes and couplings between bendings and rotations. The expectation value of these terms in the ground state IW,ai /V2,a2 > is zero. In order to study the contribution of the rotational terms, one must use more elaborate methods, as discussed in Section 7.17. The rotational terms contribute in lowest l/N order a rotational energy... 

However, all three particles in this equation are fermions with intrinsic spins S = jh. Therefore, we cannot balance the angular momentum in the reaction as written. The spins of the proton and the electron can be coupled to 0 or 1 ft and can also have relative angular momenta with any integral value from the emission process. This simple spin algebra will never yield the half-integral value on the left-hand side of the equation. Another fermion must be present among the products. 

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